An efficient numerical flux method for the solution of the stochastic collection equation

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J. A e r o s o l Sci. Vol. 28, Suppl. 1, pp. $745-$746, 1997 © 1997 Published by Elsevier Science Ltd. All rights reserved

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An Efficient Numerical Flux Method for the Solution of the Stochastic Collection Equation Andreas Bott, Institut ffir Physik der Atmosphgre Johannes Gutenberg-Universitgt Mainz, D-55099 Mainz, Germany

Keywords: Aerosol processing by clouds, impaction scavenging, nucleation scavening Introduction Clouds are very important sources or sinks for aerosol particles because (i) the nucleation an impaction scavenging reduces the aerosol number concentration (ii) collision and coalescence of cloud droplets and their subsequent evaporation yield a further depletion of the aerosol number concentration thereby producing new larger aerosol particles. Analytically the collision of two particles is treated as a stochastic process and is described by the stochastic collection equation (SCE). Many different methods have been developed for the numerical solution of the SCE. Due to its high numerical accuracy the method of Berry and Reinhardt (1974) (BRM) has become very popular. However, by construction BRM is not mass conservative. In addition, it is computationally relatively time consuming. These are two serious shortcomings when the scheme is utilized to describe the aerosol processing in a dynamical cloud model. In order to avoid these disadvantages of BRM, in the present paper a new flux method is proposed for the numerical solution of the SCE. The new scheme is exactly mass conservative and computationally very efficient.

Model Description The stochastic collection equation is given by (Pruppacher and Klett, 1997) at

-

n(x~, t ) K ( x ¢ l x ' ) n ( x ' , t)dx' o

n(x, t ) K ( x l x ' ) n ( x ' , t)dx 1

(1)

o

with n ( x , t ) is the drop number density function at time t, K ( x c l x ' ) is the collection kernel describing the rate at which a drop of mass xc = x - x ~ is collected by a drop of mass x ~ thus forming a drop of mass x. x0 is the mass of the smallest drop being involved in the collection process and xl = x / 2 . Introducing the mass density function g(y, t) dy = x n(x, t) dx with y = in r and r is the radius of drops with mass x yields the $CE as Og(y,t) f v ul x z Ot = o =~"~g(yct)'g(YclY)'g(y"t)dY-'xcx

fu ~ , t , K ( y l y l ) o g[Y, ) ~ g t Y ,

, i t,d ~ ) Y

(2)

The first integral on the right hand side of (1) and (2) describes the gain rate of drops of mass x by collision and coalescence of two smaller drops while the second integral denotes the loss of drops with mass x due to collection by other drops.

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Abstracts of the 1997 European AerosolConference

In discretized form the collision of drops with mass x~ with drops of mass xj yields a change in the mass density distributions g~, gj

gi(ilj) = gi - gi

~(ilj) xj

gj A y A t ,

g~(jli) --- gj - gj - - ~

giAyAt

(3)

Here, gi,gj are the mass density functions at grid points i , j before the collision process while gi(ilj ) and gj(jli) represent the new mass density distributions after the collision. -K(ilj ) is an average value of the collection kernel which has been obtained by two-dimensional linear interpolation. Due to the collision of drops in grid box i with drops in grid box j new drops with mass x~(ilj) = xi + xj are produced

g, (ilj ) = x'(ilj) g ( ~ ( i l j ) g ~ A y A t xixj

(4)

with xk <_ x~ + xj <_ xu+l. g'(i]j) is split up in grid boxes k and k + 1 in the following two step procedure: First g'(i]j) is entirely added to grid box k yielding

g~(ilj) = 9k + 9'(ilj)

(5)

In the second step a certain fraction of the new mass gtk(ilj ) is transported into grid box k + 1. This transport is calculated by means of an advection process through the boundary k + 1/2 between grid box k and k + 1.

gk(ilJ) = g~(ilJ) - fk+l/2(ilJ),

g~+l(ilj) -~ gk+l + h+l/2(i]j) _ ck(ilj)] fk+l/2(ilJ) = g~(ilj)[1 - ck(ilj)] + [gk+l - g~(ilj)]

ck(ilJ)[12

with ck(i[j) =

(x'(i[j) - xk)/(xk+~-

(6)

x~)g'(ilj)/g~(ilj ).

Numerical Results

In different numerical test studies the results of the flux method were compared with BRM. The agreement between both methods was generally very good. Numerical sensitivity studies were performed showing that the flux method behaves numerically stable for different choices of the numerical grid as well as different values of the integration timestep. For 0.1 _< At _< 10 s the curves were very similar thus allowing a relatively large timestep for the numerical solution of the SCE. The exact mass conservation as well as the numerical stability and efficiency of the flux method are the most important advantages when the model is compared with BRM. Therefore, this method is a very attractive alternative to BRM for solving the SCE in dynamic cloud models dealing with explicit microphysics. References

Berry, E.X, and R.L. Reinhardt, 1974: An analysis of cloud droplet growth by collection: Part I. Double distributions. J. Atmos. Sci., 31, 1814-1824. Pruppacher, H.R., and J.D. Klett, 1997: Microphysics of Clouds and Precipitation. Kluwer Academic Publishers, Dordrecht, Holland, 954 pp.